Find the Lebesgue Measure of the following sets :
$A=\{0<x\leq 1:x\sin (\dfrac{1}{x})\geq 0\}$
$B=\{0<x\leq 1:\sin(\dfrac{1}{x})\geq 0\}$
In order to find Lebesgue Measure of a set we have to cover the set by open balls in $\mathbb R^2$ and take the infimum of the area of the balls covered.
But how to cover the graph of the above sets by the open balls?
Any help will be great.
Best Answer
Some initial comments:
Now you have $$B=\{0 < x \le 1\ : \ \sin \frac{1}{x} \ge 0\} =\bigcup_{k \ge 1} [\frac{1}{2k \pi +\pi},\frac{1}{2k \pi}].$$
As those intervals are disjoint, you can find the Lebesgue measure and get $$\mu(B)=\sum_{k \ge 1} (\frac{1}{2k \pi}-\frac{1}{2k \pi +\pi})=\frac{1}{\pi}\sum_{k \ge 1} (\frac{1}{2k}-\frac{1}{2k+1})$$